Question about the definition of axisymmetric vector fields

Question

i just started to study axisymmetric vector fields. In the definition it says a vector field $V$ in cylindrical coordinates $V(\rho,\varphi,z)$ is axisymmetric if it does not depend on the angular, i.e. we actually have $V(\rho,z)$. Since we have the standard basis $e_{\rho}=(\cos(\varphi),\sin(\varphi),0)$, $e_{\varphi}=(-\sin(\varphi),\cos(\varphi),0)$ and $e_z=(0,0,1)$, \begin{align} V=V_{\rho} e_{\rho}+ V_{\varphi}e_{\varphi}+V_z e_z \end{align} So $V$ does in general depend on the angular $\varphi$. Can somebody resolve my confusion?